Stationary solutions of the equations for nonisothermal electroconvection of a weakly conducting incompressible polymeric liquid

Spectral asymptotics of a linearized problem for an incompressible weakly conducting polymeric fluid

“…A mathematical model describing nonisothermic flow of an incompressible weakly conducting polymeric fluid with a negative volume charge was proposed in [3] (for the description of polymeric media one used a new rheological model, see [1]). The formulation of the model in [3] was based, in particular, on the results from [8][9][10] obtained for an inviscid incompressible fluid. In the Cartesian coordinate system (x, y) and in a dimensionless form (the process of obtaining this form is described in detail in [3]) this mathematical model can be written as follows:…”

“…The formulation of the model in [3] was based, in particular, on the results from [8][9][10] obtained for an inviscid incompressible fluid. In the Cartesian coordinate system (x, y) and in a dimensionless form (the process of obtaining this form is described in detail in [3]) this mathematical model can be written as follows:…”

“…Ga = Ra/Pr is the Grashof number, andĒ A = E A /T 0 . Moreover, the Reynolds number Re, the Weissenberg number W , the heat conductivity coefficient σ , the mobility coefficient b, the Rayleigh number Ra, the Prandtl number Pr and the activation energy E A are constants and were described in detail in [3][4][5][6][7][8][9]. The independent and dependant variables t, x, y, u, v, P , a 11 , a 12 , a 22 , q, and in system (1)-(9) are scaled to l/u H , l, u H , ρu 2 H , W/3, (ε 0 εV 0 )/l 2 , and V 0 , respectively, where l is a characteristic length, u H is a characteristic velocity, ρ(= const) is the density, ε 0 is the electric constant, ε is the dielectric penetration, and the parameter V 0 is described below.…”

“…Stationary solutions of the electrohydrodynamic equations describing nonisothermic flows of an incompressible polymeric fluid with a volume charge were studied in [3]. These stationary solutions are counterparts of the Poiseuille and Couette flows for the Navier-Stocks equations.…”

“…In this work we study the equations obtained by the linearization of the system from [3] with respect to a basic solution describing stationary counterparts of the Poiseuille flows. Our main result is a formula describing the asymptotics of the spectrum of the linearized problem.…”